Recently, a new technique for analysing the speed of convergence of the Markov chain called
Glauber dynamics has been introduced in [1]}. This technique is called the
Spectral Independence method. The authors in [1]} use the Spectral
Independence method (SI) to prove a long standing conjecture about the mixing time of
Glauber dynamics for the so-called Hard-core model, improving on a series of result
such as [3]}, [4]}. Since then, it is not an exaggeration
to claim that SI has revolutionised the study in the field. Using this method it has been
possible to get positive results for approximate sampling from 2-spin Gibbs distributions
that match the hardness ones, e.g., [1]}, [6]}, [7]}, [8]}.
